Projection and Convolution Operations for Integrally Convex Functions

TL;DR

This paper investigates the stability of integrally convex functions under projection and convolution operations, revealing conditions under which these functions preserve their convexity properties in discrete convex analysis.

Contribution

It demonstrates that integrally convex functions are stable under projection and convolution with separable convex functions, and clarifies limitations with subclasses like discrete midpoint convexity.

Findings

Integrally convex functions are stable under projection.

Convolution with separable convex functions preserves integrally convexity.

Counterexamples show limitations for subclasses with discrete midpoint convexity.

Abstract

This paper considers projection and convolution operations for integrally convex functions, which constitute a fundamental function class in discrete convex analysis. It is shown that the class of integrally convex functions is stable under projection, and this is also the case with the subclasses of integrally convex functions satisfying local or global discrete midpoint convexity. As is known in the literature, the convolution of two integrally convex functions may possibly fail to be integrally convex. We show that the convolution of an integrally convex function with a separable convex function remains integrally convex. We also point out in terms of examples that the similar statement is false for integrally convex functions with local or global discrete midpoint convexity.